Half-trek criterion for generic identifiability of linear structural equation models

Foygel, Rina; Draisma, Jan; Drton, Mathias

doi:10.1214/12-aos1012

Cited by 52 publications

(135 citation statements)

References 16 publications

Supporting

Mentioning

134

Contrasting

Order By: Relevance

“…The concept of a hidden common cause is similar to a system of treks , which induce latent correlation; see, for example, Foygel et al (). The difference is that treks only consider pairwise dependence, not dependence among an arbitrary collection of variables.Example Let

scriptG

be the DAG in Fig.…”

Section: Mdagsmentioning

confidence: 99%

Graphs for Margins of Bayesian Networks

Evans¹

2015

Scandinavian J Statistics

View full text Add to dashboard Cite

Directed acyclic graph (DAG) models, also called Bayesian networks, impose conditional independence constraints on a multivariate probability distribution, and are widely used in probabilistic reasoning, machine learning and causal inference. If latent variables are included in such a model, then the set of possible marginal distributions over the remaining (observed) variables is generally complex, and not represented by any DAG. Larger classes of mixed graphical models, which use multiple edge types, have been introduced to overcome this; however, these classes do not represent all the models which can arise as margins of DAGs. In this paper we show that this is because ordinary mixed graphs are fundamentally insufficiently rich to capture the variety of marginal models.We introduce a new class of hyper-graphs, called mDAGs, and a latent projection operation to obtain an mDAG from the margin of a DAG. We show that each distinct marginal of a DAG model is represented by at least one mDAG, and provide graphical results towards characterizing when two such marginal models are the same. Finally we show that mDAGs correctly capture the marginal structure of causally-interpreted DAGs under interventions on the observed variables.

show abstract

scriptG

be the DAG in Fig.…”

Section: Mdagsmentioning

confidence: 99%

Graphs for Margins of Bayesian Networks

Evans¹

2015

Scandinavian J Statistics

View full text Add to dashboard Cite

show abstract

“…Finally we are in a position to easily prove Lemma 3.7. Note that, by Lemma A.5 we have that |Γ The proof of this lemma follows almost identically as the proof of Lemma 2 in Foygel et al (2012a). We simply restate the arguments there in our setting.…”

Section: Resultsmentioning

confidence: 87%

“…As our focus will be on the identification of individual edges in G we do not state the identifiability result of Foygel et al (2012a) in its usual form, instead we present a slightly modified version which is easily seen to be implied by the proof of Theorem 1 in Foygel et al (2012a). The sufficient condition for rational identifiability of G in Foygel et al (2012a) is obtained through iterative application of Theorem 2.10.…”

Section: Preliminariesmentioning

confidence: 99%

Determinantal Generalizations of Instrumental Variables

Weihs

Robinson

Dufresne

et al. 2017

Journal of Causal Inference

Self Cite

View full text Add to dashboard Cite

Abstract. Linear structural equation models relate the components of a random vector using linear interdependencies and Gaussian noise. Each such model can be naturally associated with a mixed graph whose vertices correspond to the components of the random vector. The graph contains directed edges that represent the linear relationships between components, and bidirected edges that encode unobserved confounding. We study the problem of generic identifiability, that is, whether a generic choice of linear and confounding effects can be uniquely recovered from the joint covariance matrix of the observed random vector. An existing combinatorial criterion for establishing generic identifiability is the half-trek criterion (HTC), which uses the existence of trek systems in the mixed graph to iteratively discover generically invertible linear equation systems in polynomial time. By focusing on edges one at a time, we establish new sufficient and necessary conditions for generic identifiability of edge effects extending those of the HTC. In particular, we show how edge coefficients can be recovered as quotients of subdeterminants of the covariance matrix, which constitutes a determinantal generalization of formulas obtained when using instrumental variables for identification.

show abstract

“…Checking this by counting monomials under the staircase means considering the leading monomials considering only the variables λ 12 , λ 23 Although Gröbner basis methods can be effective, it is desirable to obtain combinatorial methods that are efficient also for large-scale problems. The half-trek criteria of [34] are state-of-the-art criteria that can be checked in time that is polynomial in the size of the vertex set of the considered graph. They provide a sufficient as well as a necessary condition for generic injectivity of φ G .…”

Section: Generic Identifiabilitymentioning

confidence: 99%

“…We merely outline the proof of the theorem; for details see [34]. Some of the arguments are further illustrated in Example 8.3.…”

Section: Generic Identifiabilitymentioning

confidence: 99%

Algebraic problems in structural equation modeling

Drton¹

Advanced Studies in Pure Mathematics

Self Cite

View full text Add to dashboard Cite

The paper gives an overview of recent advances in structural equation modeling. A structural equation model is a multivariate statistical model that is determined by a mixed graph, also known as a path diagram. Our focus is on the covariance matrices of linear structural equation models. In the linear case, each covariance is a rational function of parameters that are associated to the edges and nodes of the graph. We statistically motivate algebraic problems concerning the rational map that parametrizes the covariance matrix. We review combinatorial tools such as the trek rule, projection to ancestral sets, and a graph decomposition due to Jin Tian. Building on these tools, we discuss advances in parameter identification, i.e., the study of (generic) injectivity of the parametrization, and explain recent results on determinantal relations among the covariances. The paper is based on lectures given at the 8th Mathematical Society of Japan Seasonal Institute.

show abstract

Half-trek criterion for generic identifiability of linear structural equation models

Cited by 52 publications

References 16 publications

Graphs for Margins of Bayesian Networks

Graphs for Margins of Bayesian Networks

Determinantal Generalizations of Instrumental Variables

Algebraic problems in structural equation modeling

Contact Info

Product

Resources

About